A polynomial invariant for a new class of phylogenetic networks

Abstract

Invariants for complicated objects such as those arising in phylogenetics, whether they are invariants as matrices, polynomials, or other mathematical structures, are important tools for distinguishing and working with such objects. In this paper, we generalize a complete polynomial invariant on trees to a class of phylogenetic networks called separable networks, which will include orchard networks. Networks are becoming increasingly important for their ability to represent reticulation events, such as hybridization, in evolutionary history. We provide a function from the space of internally multi-labelled phylogenetic networks, a more generic graph structure than phylogenetic networks where the reticulations are also labelled, to a polynomial ring. We prove that the separability condition allows us to characterize, via the polynomial, the phylogenetic networks with the same number of leaves and same number of reticulations by considering their internally labelled versions. While the invariant for trees is a polynomial in Z [ x 1 , … , x n , y ] where n is the number of leaves, the invariant for internally multi-labelled phylogenetic networks is an element of Z [ x 1 , … , x n , λ 1 , … , λ r , y ], where r is the number of reticulations in the network. When the networks are considered without leaf labels the number of variables reduces to r + 2.

Polynomial Invariants for Cactuses

“…In this section we modify the technique used in [17] to obtain a complete polynomial invariant for a special subclass of rooted cactuses. To define this polynomial we first require some additional notation.…”

“…Recently there has been interest in defining polynomial invariants for graphs that arise in the field of phy-Email addresses: l.j.j.vaniersel@tudelft.nl (Leo van Iersel), v.moulton@uea.ac.uk (Vincent Moulton), y.murakami@tudelft.nl (Yukihiro Murakami) logenetics [13,17]. Such graphs are called phylogenetic networks, and they often come equipped with a leaf-labelling of the vertices corresponding to some collection of species (see e.g.…”

“…More recently, polynomial invariants have also been introduced for classes of phylogenetic networks, building on Liu's approach. In [13], Liu showed how to extend the polynomial invariant B to leaf-labelled trees, and more recently a polynomial invariant was introduced for rooted binary internally multi-labelled phylogenetic networks (where vertices with indegree at least 2, or reticulations, are distinctly labelled) in [17]. This was shown to be a complete invariant for a certain subclass of such partly-labelled networks.…”

“…Our first polynomial invariant F given in Section 3 is based on unfolding a network, a technique used in [17] and inspired by [9]. When unfolding a rooted cactus, copies of the subnetwork rooted at every reticulation are created.…”

Quantifying syntax similarity with a polynomial representation of dependency trees

“…This builds an one-to-one correspondence between unlabeled trees and a class of bivariate polynomials, that is, two unlabeled trees are isomorphic if and only if they have the same polynomial. This tree distinguishing polynomial has been applied to study phylogenetic trees and pathogen evolution (P. Liu et al, 2022) and generalized to represent some classes of phylogenetic networks (Janssen and Liu, 2021;Pons et al, 2022;van Iersel et al, 2022).…”